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Open Problems


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Random stable roommatesMertens✭✭0Graph Theory » Basic G.T. » Matchingsmdevos
Rank vs. GenusJohnson✭✭✭0TopologyJesse Johnson
Real roots of the flow polynomialWelsh✭✭0Graph Theory » Coloring » Nowhere-zero flowsmdevos
Realisation problem for the space of knots in the 3-sphereBudney✭✭0Topologyrybu
Reconstruction conjectureKelly; Ulam✭✭✭✭0Graph Theoryzitterbewegung
Reed's omega, delta, and chi conjectureReed✭✭✭0Graph Theory » Coloring » Vertex coloringmdevos
Refuting random 3SAT-instances on $O(n)$ clauses (weak form)Feige✭✭✭0Theoretical Comp. Sci. » Complexity » Hardness of Approximationcwenner
Rendezvous on a lineAlpern✭✭✭0Unsortedmdevos
Roller Coaster permutationsAhmed; Snevily✭✭✭0CombinatoricsTanbir Ahmed
Rota's basis conjectureRota✭✭✭0Combinatorics » Matricesmdevos
Rota's unimodal conjectureRota✭✭✭0Combinatorics » Matroid Theorymdevos
Ryser's conjectureRyser✭✭✭0Graph Theory » Hypergraphsmdevos
S(S(f)) = S(f) for reloidsPorton✭✭0Topologyporton
Saturated $k$-Sperner Systems of Minimum SizeMorrison; Noel; Scott✭✭1Combinatorics » PosetsJon Noel
Saturation in the HypercubeMorrison; Noel; Scott✭✭0CombinatoricsJon Noel
Schanuel's ConjectureSchanuel✭✭✭✭0Number Theory » Analytic N.T.Charles
Seagull problemSeymour✭✭✭0Graph Theory » Basic G.T. » Minorsmdevos
Sequence defined on multisetsErickson✭✭1CombinatoricsMartin Erickson
Sets with distinct subset sumsErdos✭✭✭0Number Theory » Combinatorial N.T.mdevos
Several ways to apply a (multivalued) multiargument function to a family of filtersPorton✭✭✭0Topologyporton
Seymour's r-graph conjectureSeymour✭✭✭0Graph Theory » Coloring » Edge coloringmdevos
Seymour's Second Neighbourhood ConjectureSeymour✭✭✭1Graph Theory » Directed Graphsnkorppi
Seymour's self-minor conjectureSeymour✭✭✭0Graph Theory » Infinite Graphsmdevos
Shannon capacity of the seven-cycle✭✭✭0Graph Theorytchow
Shuffle-Exchange ConjectureBeneš; Folklore; Stone✭✭✭0CombinatoricsVadim Lioubimov
Shuffle-Exchange Conjecture (graph-theoretic form)Beneš; Folklore; Stone✭✭✭0Graph TheoryVadim Lioubimov
Sidorenko's ConjectureSidorenko✭✭✭0Graph TheoryJon Noel
Signing a graph to have small magnitude eigenvaluesBilu; Linial✭✭0Graph Theorymdevos
Simplexity of the n-cube✭✭✭1Geometrymdevos
Simultaneous partition of hypergraphsKühn; Osthus✭✭0Graph Theory » Hypergraphsfhavet
Singmaster's conjectureSingmaster✭✭1Number Theory » Combinatorial N.T.Zach Teitler
Slice-ribbon problemFox✭✭✭✭0Topologyrybu
Smooth 4-dimensional Poincare conjecturePoincare; Smale; Stallings✭✭✭✭0Topologyrybu
Smooth 4-dimensional Schoenflies problemAlexander✭✭✭✭0Topologyrybu
Snevily's conjectureSnevily✭✭✭1Number Theory » Combinatorial N.T.mdevos
Something like Picard for 1-formsElsner✭✭0AnalysisMathOMan
Special PrimesGeorge BALAN✭1Number Theorymaththebalans
Splitting a digraph with minimum outdegree constraintsAlon✭✭✭0Graph Theory » Directed Graphsfhavet
Square achievement game on an n x n gridErickson✭✭1CombinatoricsMartin Erickson
Stable set meeting all longest directed paths.Laborde; Payan; Xuong N.H.✭✭0Graph Theoryfhavet
Star chromatic index of complete graphsDvorak; Mohar; Samal✭✭1Graph TheoryRobert Samal
Star chromatic index of cubic graphsDvorak; Mohar; Samal✭✭0Graph TheoryRobert Samal
Sticky Cantor sets✭✭0Topologyporton
Strict inequalities for products of filtersPorton✭0Topologyporton
Strong 5-cycle double cover conjectureArthur; Hoffmann-Ostenhof✭✭✭1Graph Theory » Basic G.T. » Cyclesarthur
Strong colorabilityAharoni; Alon; Haxell✭✭✭0Graph Theory » Coloring » Vertex coloringberger
Strong edge colouring conjectureErdos; Nesetril✭✭0Graph Theory » Coloring » Edge coloringfhavet
Strong matchings and coversAharoni✭✭✭0Graph Theory » Infinite Graphsmdevos
Sub-atomic product of funcoids is a categorical product✭✭0Algebraporton
Subdivision of a transitive tournament in digraphs with large outdegree. Mader✭✭0Graph Theory » Directed Graphsfhavet

Imp.¹: Importance (Low ✭, Medium ✭✭, High ✭✭✭, Outstanding ✭✭✭✭)
Rec.²: Recommended for undergraduates.

Note: Resolved problems from this section may be found in Solved problems.

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